1. Gödel never proved the theorem known today as "Gödel's Second Incompleteness Theorem" (he proved Gödel's First Incompleteness Theorem) - he only announced this Theorem as a hypothesis and he promissed to prove it, but he never did it. The first proof of this Theorem was published in Hilbert and Bernays' monograph "Grudnlagen der Mathematik" in 1939 (for e.g.: R. Murawski, "Recursive Functions and Metamathematics", Springer Science+Business Media Dordrecht 1999).
2. referring to the issue of provability consistency of Arithmetic System, I would like to inform you about the paper: T. J. Stępień, Ł. T. Stępień, „On the Consistency of the Arithmetic System”, Journal of Mathematics and System Science, vol. 7, 43 (2017), arXiv:1803.11072.
There in this paper, was published a proof of the consistency of the Arithmetic System. This proof had been done within this Arithmetic System (the abstract related to this paper: T. J. Stepien and L. T. Stepien, "On the consistency of Peano's Arithmetic System" , The Bulletin of Symbolic Logic, vol. 16, No. 1, 132 (2010)).
"but if you accept that the halting problem is unlovable" I think you meant unsolvable here? Thank you! I enjoyed reading the article!
I have two comments here:
1. Gödel never proved the theorem known today as "Gödel's Second Incompleteness Theorem" (he proved Gödel's First Incompleteness Theorem) - he only announced this Theorem as a hypothesis and he promissed to prove it, but he never did it. The first proof of this Theorem was published in Hilbert and Bernays' monograph "Grudnlagen der Mathematik" in 1939 (for e.g.: R. Murawski, "Recursive Functions and Metamathematics", Springer Science+Business Media Dordrecht 1999).
2. referring to the issue of provability consistency of Arithmetic System, I would like to inform you about the paper: T. J. Stępień, Ł. T. Stępień, „On the Consistency of the Arithmetic System”, Journal of Mathematics and System Science, vol. 7, 43 (2017), arXiv:1803.11072.
There in this paper, was published a proof of the consistency of the Arithmetic System. This proof had been done within this Arithmetic System (the abstract related to this paper: T. J. Stepien and L. T. Stepien, "On the consistency of Peano's Arithmetic System" , The Bulletin of Symbolic Logic, vol. 16, No. 1, 132 (2010)).